I tried searching but it's not working on my computer for some reason, so sorry if this was already asked.

I have some teaching experience through programs at my college, but I don't have any documents from the departments saying that I did work there. However, they said I could just put their contact information as references. I'm thinking it's better to just get something signed in paper from them stating that I did work there for such and such period of time.

If you want the experience to be counted towards your pay scale, you must present a letter/form from the employer... on their letterhead... signed... preferably with some sort of fancy stamp or seal. It will list at the least the basics of your experience there... dates... duties... salary (sometimes)... and that you left in good standing.

Do not confuse this with references. They are two different things although they can be the same people. This is not something done in the US, but often you can convince your current employer and perhaps some past ones to create this simple form.